A common conversation I have with students is about interpreting p values (normally given by SPSS). P values are about probability, an area of mathematics I find particularly interesting.

Back when I wore a younger man’s clothes, a trainee maths teacher, who was observing an A level mathematics lesson I was giving, asked me “do you believe in probability?” I did not really understand the question and by the look in the faces of the teenagers in front of me, neither did they. I asked for clarification and was met with “well do you think it actually exists, or is it just luck?” To this day, I never fully understood the question, though it does highlight just how tricky the concepts of probability and statistics are.

I rather enjoy the quirkiness that comes with the study of probability. In secondary school, probability involved rolling dice and writing fractions on trees. Someone once told me that the probability of anything happening was always 50-50 - because either it does happen or it does not! The probability of an event is a numerical value between 0 and 1 (or 100%) that tells us the likeliness that said event will happen. Theoretical probabilities can be calculated as a fraction by writing the number of ways an outcome can happen divided by the total number of outcomes there are. The probability of rolling a 3 on a fair die is ⅙ since there is a single 3 out of 6 possible outcomes. Although it isn’t always 50% - 50%, the total probability of all possible outcomes of an event will always add up to 1 (100%).

Two of my favorite probability curiosities:

1

With players plus referee, there are 23 people on a football field (2 teams of 11 plus 1 referee). With 23 people there is a better than 50% chance that two of them share a birthday. If you include assistant referees, managers, coaching staff and substitutes the probability of two people sharing a birthday becomes really quite high (above 70% for a group of 30 people). To demonstrate this we employ a mathematical trick of looking at the question from a different perspective. It is shown by working out the probability that no one shares a birthday in a group. So in a group of 2 people (not sure we can call 2 people a group) the probability that they don’t share a birthday is 364/365 (there are 365 days in a year, 364 are different birthdays). The probability that they do share a birthday is 1 minus this answer, they either do or they don’t, so it’s 1/365 (0.3%). For 3 people the probability that they don’t share a birthday is 364/365 multiplied by 363/365 (there are 364 different days for the 2nd person and 363 different days for the 3rd person, to work out probability of both happening we multiply the probabilities). Again, take this answer from 1 to get the probability of sharing a birthday, which comes out as 0.8%. Continuing this principle, we get a probability of more than 50% at as little as 23 people.

2

There are 52 playing cards in a standard deck. The number of different ways that these can be ordered is 52 x 51 x 50 x …. x 3 x 2 x 1 (there are 52 places for the first card to go, then 51 for the second and so on). This number is called 52 factorial and written 52!. The number itself is massive, bigger than 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. There are this many ways that a deck of cards can be ordered. To put it into context the number 8 billion (about the population of the Earth) is 8,000,000,000. Therefore, when you shuffle a pack of cards the chance that you get an order of cards that has been seen before, anywhere on the planet, is extremely close to zero. So go home, shuffle a pack of cards; you would have boldly gone where no one has gone before - as no one else would have ever put them in that order.

So p-values. Many students make use of statistical software (such as SPSS) and are asked to look for significance and find a p value. Often students are trying to show that two groups are different (for example a control and an experimental group). This is done by analysing samples taken from the groups and exploring differences between calculated statistics (the mean for example). The software programme does some computations and gives the user a p value. We are told that we reject the null hypothesis if the p value is less than 0.05 (5%). What is the null hypothesis, what is this p value telling us and why do we reject the null hypothesis (and find significance) when it’s less than a certain value?

The null hypothesis is the ‘no difference’ conclusion. It is what we are testing our groups against. So if we were exploring if there is a difference between the control and experimental group our null hypothesis would be that there is not a difference. A statistical test is carried out to compare the groups, based on collected data, and the p value is calculated. Opposed to the null hypothesis is the alternative hypothesis. This states that there is a difference between the groups and we go to this, if we can reject the null hypothesis.

If we reject the null hypothesis, in favour of the alternative, there is a chance that we have made a mistake (in statistics we call it a type I error). In a criminal trial, we can think of the null hypothesis as ‘not guilty’ and we test the facts against this, deciding whether to reject it in favour of the alternative, namely ‘guilty’. Sometimes innocent people are wrongly found guilty, this would be a type I error (not much comfort when you’re serving time). In statistics, the probability that we make a type I error and so wrongly reject the null hypothesis, is this p value. For us to be confident in finding the difference between the two groups to be significant, we would want this p value to be small (a low probability of making this type I error). A common level is 0.05 (5%). So if we were to get a p value of 0.021 (2.1%) this would mean that the probability of us rejecting the null hypothesis, of no difference, incorrectly is only 2.1%. We can therefore say there is a significant difference.

Carrying out statistical tests on samples of data is a component of many degree courses. Some examples where you would carry out these tests are bioscience, psychology and sports science. If you would like to know more about carrying out statistical tests then please look out for **statistics workshops** or book a **1 to 1 with me.**

Phil RobertsMaths & Statistics Adviser |

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